Junction-tree sum-product message ($m_{i\to j}(x_{S_{ij}})={\sum }_{x_{C_i\setminus S_{ij}}}{\psi }_i(x_{C_i}){\prod }_{k\sim i,k\ne j}{m}_{k\to i}$)

Assign each factor exactly once to a containing clique and multiply the factors there into ${\psi }_i$. The displayed sum-product belief propagation message sums out variables of $C_i$ outside the separator $S_{ij}=C_i\cap C_j$. By the running intersection property, eliminating all variables in the subtree on the $i$ side creates precisely this function of the separator. An inward pass computes the normalizing constant at the root; an outward pass produces marginal distributions. For pedigree likelihoods, observed genotypes and penetrance factors simply multiply the relevant potentials.